\begin{frame}[allowframebreaks]
\frametitle{Labelled table matrix grammars}

\begin{define}[labelled TXMG]
For $X \in \{R, CF, CS\}$ $G = (G_1, L, L^{ac})$ is a labelled TXMG where
\begin{itemize}
	\item $G_1 = (G_H, G_V)$ is a TXMG with $G_H$ and $G_V = (\cup_{i = 1}^{k}G_i, \mathcal{P})$ as usual. 
	\item L = a finite set of labels for tables in $\mathcal{P}$. 
	\item $L^{ac} \subseteq L$
\end{itemize}
\end{define}

\begin{define}
$N = \cup_{i = 1}^{k} N_i$ are the set of all nonterminals. 
\begin{enumerate}
	\item If t is table of $\mathcal{P}$, then $reg(t) = \{A \in N \vert A \rightarrow \alpha \text{ is a rule in t}\}$
	\item If a Matrix M is generated with vertical derivation of a TXMG and M is as follows: 
	\[
	M = 
	\boxed{
	\begin{aligned}
	\begin{matrix}
	a_{11} & \dots & a_{1n} \\[-1ex]
	\vdots & \vdots & \vdots \\[-1ex]
	a_{r1} & \dots & a_{rn} \\[-0.5ex]
	A_1 & \dots & A_n
	\end{matrix}
	\end{aligned}
	}
	\]
	with $a_{ij} \in T, A_j \in N, i = 1, \dots, r; j = 1, \dots, n$
	
	\ \\
	
	then $min_N(M) = \{A \in N \vert \text{A occurs in M}\}$
\end{enumerate}
\end{define}

\end{frame}

\begin{frame}[allowframebreaks]
\frametitle{Derivation}

Let $N = \cup_{i = 1}^{k} N_i$ and M, M' two matrixes with 

\[
M = 
\boxed{
\begin{aligned}
\begin{matrix}
a_{11} & \dots & a_{1n} \\[-1ex]
\vdots & \vdots & \vdots \\[-1ex]
a_{r1} & \dots & a_{rn} \\[-0.5ex]
A_1 & \dots & A_n
\end{matrix}
\end{aligned}
}
\]

We write: 

\begin{itemize}
	\item $M \overset{l}{\underset{G_V}{\Downarrow}} M'$, if M' is generated from M over $G_V$ using table t with label l. 
	\item $M \overset{l}{\underset{G_V}{\Downarrow_{mt}}} M'$, if the condition above is satisfied and $reg(t) = min_N(M)$. 
	\item $M \overset{*}{\underset{G_V}{\Downarrow}} M'(z)$, if there exists a control word $z = l_1 \dots l_d$ with $l_i \in L, i = 1, \dots, d$. 
	\item $M \overset{*}{\underset{G_V}{\Downarrow^{ac}}} M'(z)$, if the condition above is satisfied and there exists matrixes $M_0, \dots, M_d$ with $M_0 = M$, $M_d = M'$ and for every $i = 1, \dots, d$
	\begin{itemize}
		\item if $min_N(M_{i - 1}) \subseteq reg(t_i)$ where $t_i$ is the table with label $l_i$, then $M_{i-1} \overset{l_i}{\underset{G_V}{\Downarrow}} M_i$
		\item otherwise, $l_i \in L^{ac} \text{ and } M_{i-1} = m_i$. 
	\end{itemize}
	\item $M \overset{*}{\underset{G_V}{\Downarrow_{mt}}} M'$ and $M \overset{*}{\underset{G_V}{\Downarrow_{mt}^{ac}}} M'$ are defined in the same way. 
\end{itemize}

\end{frame}

\begin{frame}
\frametitle{TXMG generated sets}

\begin{define}
The set of matrices generated by a labelled TXMG G is

$M_i^j(G) = \{M \vert S \overset{*}{\underset{G_H}{\Rightarrow}} S_1 \dots S_n \overset{*}{\underset{G_V}{\Downarrow_i^j}} M \in T^{**}, S_i \text{ is intermediate in } G_H\}$

where i may be mt or not and j may be ac or not. 

We call $M_i^j(G)$ a $TXML_i^j$. 
\end{define}

\end{frame}

\begin{frame}[allowframebreaks]
\frametitle{Labelled TXMG with Y control}

\begin{define}[(Y)TXMG]
$G' = (G, C)$ is a labelled TXMG with Y control (for $X, Y \in \{R, CF, CS\}$) where
\begin{itemize}
	\item G is a labelled TXMG
	\item C is a Y control language over the label set L of G. 
\end{itemize} 
\end{define}

\begin{define}
The set of matrices generated by a (Y)TXMG G' is
\begin{align*}
M_i^j(G') = M_i^j(G, C) = \{M \vert S \overset{*}{\underset{G_H}{\Rightarrow}} S_1 \dots S_n \overset{*}{\underset{G_V}{\Downarrow_i^j}} M(z) \in T^{**}, \\
S_i \text{ is intermediate in } G_H, z \in C\}
\end{align*}

where i may be mt or not and j may be ac or not. 

\end{define}

We call a Y controlled $M_i^j(G')$ a $(Y)TXML_i^j$. 

\end{frame}

\begin{frame}[allowframebreaks]
\frametitle{Example}

\begin{Example}
$G' = (G, C)$ is a (CS)TCSMG with intermediates $\{S_1, S_2, S_3\}$ and $G = (G_H, G_V)$ where
\begin{itemize}
	\item $L(G_H) = \{S_1^nS_2^nS_3^n \vert n \geq 1\}$
	\item the tables for $G_V$ are:
	\begin{itemize}
		\item $t_1 = \{S_1 \rightarrow .S_1, S_2 \rightarrow xS_2, S_3 \rightarrow +S_3\}$
		\item $t_2 = \{S_1 \rightarrow xS_1, S_2 \rightarrow xS_2, S_3 \rightarrow +S_3\}$
		\item $t_3 = \{S_1 \rightarrow +S_1, S_2 \rightarrow +S_2, S_3 \rightarrow +S_3\}$
		\item $t_4 = \{S_1 \rightarrow +, S_2 \rightarrow +, S_3 \rightarrow +\}$
	\end{itemize}
	\item $C = \{t_1^mt_2^mt_3^mt_4 \vert m \geq 1\}$
\end{itemize}
\end{Example}

\[
\boxed{
\begin{aligned}
\begin{matrix}
S_1 & S_2 & S_3
\end{matrix}
\end{aligned}
}
\overset{t_1}{\underset{G_V}{\Downarrow}}
\boxed{
\begin{aligned}
\begin{matrix}
. & x & + \\[-0.5ex]
S_1 & S_2 & S_3
\end{matrix}
\end{aligned}
}
\overset{t_2}{\underset{G_V}{\Downarrow}}
\boxed{
\begin{aligned}
\begin{matrix}
. & x & + \\[-0.5ex]
x & x & + \\[-0.5ex]
S_1 & S_2 & S_3
\end{matrix}
\end{aligned}
}
\]

\[
\overset{t_3}{\underset{G_V}{\Downarrow}}
\boxed{
\begin{aligned}
\begin{matrix}
. & x & + \\[-0.5ex]
x & x & + \\[-0.5ex]
+ & + & + \\[-0.5ex]
S_1 & S_2 & S_3
\end{matrix}
\end{aligned}
}
\overset{t_4}{\underset{G_V}{\Downarrow}}
\boxed{
\begin{aligned}
\begin{matrix}
. & x & + \\[-0.5ex]
x & x & + \\[-0.5ex]
+ & + & + \\[-0.5ex]
+ & + & +
\end{matrix}
\end{aligned}
}
\]

\end{frame}

\begin{frame}

\begin{thm}
\begin{align*}
TXML = TXML^{ac} = (R)TXML \\
\subseteq TXML_{mt} = TXML_{mt}^{ac} = (R)TXML^{ac} = (R)TXML_{mt} = (R)TXML_{mt}^{ac}
\end{align*}
\end{thm}

\begin{proof}
see \cite{sironmoney1977parallelsequential}. 
\end{proof}

\end{frame}

\begin{frame}
\frametitle{Further work}

\begin{itemize}
	\item There are grammars using control languages without labels. 
	\item What happens, if Szilard language properties are applied on 2d languages. 
\end{itemize}

\end{frame}